Rates and Returns – Quantitative Methods, CFA® Level I

2. Nominal, Effective Annual Rate (EAR), and Continuous Compounding

Many exam questions test your ability to move between a quoted nominal rate, the effective annual rate (EAR), and a continuously compounded rate.

2.1 Nominal Rate

The nominal rate is the stated annual interest rate, typically quoted together with a compounding frequency:

2.2 Effective Annual Rate (EAR) Formula

The effective annual rate (EAR) is the actual annual interest rate you earn or pay after taking intra-year compounding into account.

Where r is the nominal annual rate and m is the number of compounding periods per year.

2.3 Continuous Compounding

Continuous compounding is the mathematical limit of compounding frequency going to infinity.

r_c is the continuously compounded rate; ln is the natural log.

3. Holding Period Return (HPR) and Annualized Return

You need to know both holding period return and annualized return formulas and when to use each.

3.1 Holding Period Return (HPR)

Holding period return measures total return over a single holding period:

P₀ = initial price, P₁ = ending price, D = income (dividends, coupons).

In words, HPR is the percentage gain or loss on an investment over a specified period, without adjusting for inflation.

3.2 Annualized Return from HPR

To compare investments with different holding periods, convert HPR to an annualized return:

n is the number of years in the holding period. If you hold an asset for 6 months, n = 0.5.

When you annualize multi-year returns using this geometric approach, you are effectively computing a compound annual growth rate (CAGR).

4. Nominal vs Real Returns (Impact of Inflation)

Nominal returns ignore inflation; real returns adjust for inflation and better reflect changes in purchasing power.

On the exam, this approximation is usually good enough unless the question explicitly requires an exact Fisher-equation calculation.

5. Rate Conversions at a Glance

You should be able to move quickly between:

• Nominal rate → EAR (using compounding frequency)
• EAR → continuously compounded rate r_c
• HPR → annualized return

Many CFA Level I questions give you rates in one form and ask you to compare products or investments using another. Practise these conversions until they are automatic.

For a more formula-dense reference, also review the CFA Level I Formula Sheet.

This is a sample page from the FinQuiz Battle Ready Summaries. Notice how:

• Each formula is boxed and color-highlighted
• Steps are broken into short bullets for fast review
• Links to other Quant topics are shown visually

Open Sample Summary PDF

6. How Rates and Returns Feed into Risk and Portfolio Evaluation

The formulas here don’t live in isolation. They feed directly into:

• Risk–return analysis (expected return, variance, standard deviation)
• Portfolio performance measures such as Sharpe ratio and Treynor ratio
• Fixed income yield measures and equity valuation models

When you calculate a return or a rate, always ask: “How will this number be used in a portfolio or valuation decision?”

Risk-free rate – quick definition

The risk-free rate is the theoretical return on an investment with no default risk and no reinvestment risk. In practice, it is usually approximated by the yield on a short-term government security.

7. Common Exam Pitfalls in Rates and Returns

• Mixing up nominal and effective rates. If compounding is involved, convert to EAR before comparing.
• Forgetting to include income in HPR. Dividends or coupons must be added to price changes.
• Using simple averages instead of geometric (annualized) returns when asked about multi-year performance.
• Ignoring the time period. Always adjust HPR to a per-year basis before comparing across investments.
• Confusing real and nominal returns when inflation is mentioned in the question stem.

8. Next Step: Lock In the Formulas with Battle Ready Summaries